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New oscillation criteria for higher order delay dynamic equations on time scales

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Abstract

In this paper, we investigate the oscillation of the following higher order delay dynamic equation: { a n ( t ) [ ( a n 1 ( t ) ( ( a 1 ( t ) x Δ ( t ) ) Δ ) Δ ) Δ ] α } Δ +g(t,x(τ(t)))=0 on any time scale T with supT=. Here n2, a k (t) C rd (T,(0,)) (1kn), τ:TT is an increasing differentiable function with τ(t)t and lim t τ(t)=, gC(T×R,R) with g(t,x)/ x β q(t) for some q(t) C rd (T,(0,)) when x0, and α1, β1 are two quotients of odd positive integers. We give sufficient conditions under which every solution of this equation is either oscillatory or tends to zero.

MSC:34K11, 34N05, 39A10.

1 Introduction

In this paper, we investigate the oscillation of the following higher order delay dynamic equation:

{ a n ( t ) [ ( a n 1 ( t ) ( ( a 1 ( t ) x Δ ( t ) ) Δ ) Δ ) Δ ] α } Δ +g ( t , x ( τ ( t ) ) ) =0
(E)

on some time scale T. Here n2, a k (t) C rd (T,(0,)) (1kn), τ:TT is an increasing differentiable function with τ(t)t and lim t τ(t)=, gC(T×R,R) with g(t,x)/ x β q(t) for some q(t) C rd (T,(0,)) when x0, and α1, β1 are two quotients of odd positive integers. Write

S k ( t , x ( t ) ) = { x ( t ) , if  k = 0 , a k ( t ) S k 1 Δ ( t , x ( t ) ) , if  1 k n 1 , a n ( t ) [ S n 1 Δ ( t , x ( t ) ) ] α , if  k = n ,

then (E) reduces to the equation

S n Δ ( t , x ( t ) ) +g ( t , x ( τ ( t ) ) ) =0.
(1.1)

Since we are interested in the oscillatory behavior of solutions near infinity, we assume that supT= and t 0 T is a constant. We define the time scale interval [ a , ) T ={tT:ta}. A nontrivial real-valued function x is said to be a solution of (1.1) if x C rd ( [ T x , ) T ,R), T x t 0 , which has the property that S k (t,x) C rd 1 ( [ T x , ) T ,R) for 1kn, and satisfies (1.1) on [ T x , ) T . The solutions vanishing in some neighborhood of infinity will be excluded from our consideration. A solution x of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is called nonoscillatory. The theory of time scales, which has recently received a lot of attention, was introduced by Stefan Hilger in [1] in order to unify continuous and discrete analysis. The cases when a time scale T is equal to R or the set of all integers Z represent the classical theories of differential and difference equations. Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their continuous counterparts. The study of dynamic equations on time scales reveals such discrepancies, and it helps avoid proving results twice-once for differential equations and once again for difference equations. The general is to prove a result for a dynamic equation where the domain of the unknown function is a time scale T. In this way results not only related to the set of real numbers or set of integers but those pertaining to more general time scales are obtained. Therefore, not only can the theory of dynamic equations unify the theories of differential equations and difference equations, but it also extends these classical cases to cases ‘in between’, e.g., to the so-called q-difference equations when T={1,q, q 2 ,, q n ,}, which has important applications in quantum theory (see [2]). In this work, knowledge and understanding of time scales and time scale notation are assumed, for an excellent introduction to the calculus on time scales; see Bohner and Peterson [3, 4]. In recent years, there has been much research activity concerning the oscillation and asymptotic behavior of solutions of some dynamic equations on time scales.

In [5], Hassan studied the third-order dynamic equation

( a ( t ) { [ r ( t ) x Δ ( t ) ] Δ } γ ) Δ +f ( t , x ( τ ( t ) ) ) =0
(1.2)

on a time scale T, where γ1 is the quotient of odd positive integers, a and r are positive rd-continuous functions on T, and the so-called delay function τ:TT satisfies τ(t)t for tT and lim t τ(t)= and fC(T×R,R) and obtained some oscillation criteria, which improved and extended the results that have been established in [68].

Li et al. in [9] also discussed the oscillation of (1.2), where γ>0 is the quotient of odd positive integers, fC(T×R,R) is assumed to satisfy uf(t,u)>0 for u0 and there exists a positive rd-continuous function p on T such that f ( t , u ) u γ p(t) for u0. They established some new sufficient conditions for the oscillation of (1.2).

Wang and Xu in [10] extended the Hille and Nehari oscillation theorems to the third-order dynamic equation

( r 2 ( t ) ( ( r 1 ( t ) x Δ ( t ) ) Δ ) γ ) Δ +q(t)f ( x ( t ) ) =0

on a time scale T, where γ1 is a ratio of odd positive integers and the functions r i (t) (i=1,2), q(t) are positive real-valued rd-continuous functions defined on T.

Erbe et al. in [11] were concerned with the oscillation of the third-order nonlinear functional dynamic equation

( a ( t ) [ ( r ( t ) x Δ ( t ) ) Δ ] γ ) Δ +f ( t , x ( g ( t ) ) ) =0

on a time scale T, where γ is the quotient of odd positive integers, a and r are positive rd-continuous functions on T, and g:TT satisfies lim t g(t)= and fC(T×R,R). The authors obtained some new oscillation criteria and extended many known results for oscillation of third-order dynamic equations.

Qi and Yu in [12] obtained some oscillation criteria for the fourth-order nonlinear delay dynamic equation

x Δ 4 (t)+p(t) x γ ( τ ( t ) ) =0

on a time scale T, where γ is the ratio of odd positive integers, p is a positive real-valued rd-continuous function defined on T, τ C rd (T,T), τ(t)t, and lim t τ(t)=.

Grace et al. in [13] were concerned with the oscillation of the fourth-order nonlinear dynamic equation

x Δ 4 (t)+q(t) x λ (t)=0
(1.3)

on a time scale T, where λ is the ratio of odd positive integers, q is a positive real-valued rd-continuous function defined on T. They reduced the problem of the oscillation of all solutions of (1.3) to the problem of oscillation of two second-order dynamic equations and gave some conditions to ensure that all bounded solutions of (1.3) are oscillatory.

Grace et al. in [14] established some new criteria for the oscillation of the fourth-order nonlinear dynamic equation

( a ( t ) x Δ 2 ( t ) ) Δ 2 +f ( t , x σ ( t ) ) =0,t t 0 ,

where a is a positive real-valued rd-continuous function satisfying t 0 σ ( s ) a ( s ) Δs<, f: [ t 0 , ) T ×RR is continuous satisfying sgnf(t,x)=sgnx and f(t,x)f(t,y) for xy and t t 0 . They also investigate the case of strongly superlinear and the case of strongly sublinear equations subject to various conditions.

Agarwal et al. in [15] were concerned with oscillatory behavior of a fourth-order half-linear delay dynamic equation with damping

( r ( t ) ( x Δ 3 ( t ) ) γ ) Δ +p(t) ( x Δ 3 ( t ) ) γ +q(t) x γ ( τ ( t ) ) =0
(1.4)

on a time scale T with supT=, where λ is the ratio of odd positive integers, r, p, q are positive real-valued rd-continuous functions defined on T, r(t)μ(t)p(t)0, τ C rd (T,T), τ(t)t and τ(t) as t. They established some new oscillation criteria of (1.4).

Zhang et al. in [16] concerned with the oscillation of a fourth-order nonlinear dynamic equation

( p ( t ) x Δ 3 ( t ) ) Δ +q(t)f ( x ( σ ( t ) ) ) =0
(1.5)

on an arbitrary time scale T with supT=, where p,q C rd (T,(0,)) with t 0 1 p ( s ) Δs< and there exists a positive constant L such that f ( y ) y L for all y0, they gave a new oscillation result of (1.5).

In [17], Sun et al. studied the following higher order dynamic equation:

S n Δ ( t , x ( t ) ) +p(t) x β (t)=0

and established some new oscillation criteria.

For much research concerning the oscillation and nonoscillation of solutions of higher order dynamic equations on time scales, please refer to the literature [1827].

2 Some lemmas

In order to obtain the main results of this paper, we need the following lemmas.

Lemma 2.1 [28]

Assume that

t 0 [ 1 a n ( s ) ] 1 α Δs= t 0 Δ s a i ( s ) =for all 1in1,
(2.1)

and integer m[1,n]. Then:

  1. (1)

    lim inf t S m (t,x(t))>0 implies lim t S i (t,x(t))= for i[0,m1].

  2. (2)

    lim sup t S m (t,x(t))<0 implies lim t S i (t,x(t))= for i[0,m1].

Lemma 2.2 [28]

Assume that (2.1) holds. If S n Δ (t,x(t))<0 and x(t)>0 for t t 0 , then there exists an integer m[0,n] such that:

  1. (1)

    m+n is even.

  2. (2)

    ( 1 ) m + i S i (t,x(t))>0 for t t 0 and i[m,n].

  3. (3)

    If m1, then there exists T t 0 such that S i (t,x(t))>0 for tT and i[1,m1].

Lemma 2.3 [28]

Assume that (2.1) holds. Furthermore, suppose that

t 0 1 a n 1 ( u ) { u [ 1 a n ( s ) s q ( v ) Δ v ] 1 α Δ s } Δu=.
(2.2)

If x is an eventually positive solution of (1.1), then there exists sufficiently large T t 0 such that:

  1. (1)

    S n Δ (t,x(t))<0 for tT.

  2. (2)

    Either lim t x(t)=0 or S i (t,x(t))>0 for tT and 0in.

Lemma 2.4 [28]

Assume that x is an eventually positive solution of (1.1). If there exists T t 0 such that:

  1. (1)

    S n Δ (t,x(t))<0 for tT.

  2. (2)

    S i (t,x(t))>0 for tT and 0in.

Then

S i ( t , x ( t ) ) S n 1 α ( t , x ( t ) ) B i + 1 (t,T)for 0in1 and tT
(2.3)

and there exist T 1 >T and a constant c>0 such that

x(t)c B 1 (t,T)for t T 1 ,
(2.4)

where

B i (t,T)= { T t [ 1 a n ( s ) ] 1 α Δ s , if  i = n , T t B i + 1 ( s , T ) a i ( s ) Δ s , if  1 i n 1 .
(2.5)

Lemma 2.5 [3]

Let f:RR be continuously differentiable and suppose that g:TR is delta differentiable. Then fg is delta differentiable and

( f g ) Δ (t)= g Δ (t) 0 1 f ( h g ( t ) + ( 1 h ) g σ ( t ) ) dh.

Lemma 2.6 [17]

If A, B are nonnegative numbers and λ>1, then

A λ λA B λ 1 +(λ1) B λ 0.

Lemma 2.7 [29]

Assume that U, V are constants and γ1 is the quotient of odd positive integers. Then

( U V ) 1 + 1 γ U 1 + 1 γ + 1 γ V 1 + 1 γ ( 1 + 1 γ V 1 γ U ) .

3 Main results

Throughout this paper, we assume that:

  1. (1)

    τσ=στ, where the forward jump operator σ:TT by σ(t)=inf{sT:s>t}.

  2. (2)

    Φ:T(0,) and ϕ:T[0,) such that Φ(t) and a(t)ϕ(t) are differentiable.

Write

h 1 ( t , T ) = B 2 ( t , T ) a 1 ( t ) , h 2 ( t , T ) = h 1 ( t , T ) B 1 α 1 ( σ ( t ) , T ) = h 1 ( t , T ) ( B 1 α 1 ( t , T ) ) σ , δ 1 ( t , T , c 1 , c 2 ) = { c 1 , c 1  are any positive constant , if  α < β , 1 , if  α = β , c 2 B 1 β α ( σ ( t ) , T ) , c 2  are any positive constant , if  α > β ,
(3.1)
δ 2 ( t , T , c 1 , c 2 ) = { c 1 , c 1  are any positive constant , if  α < β , 1 , if  α = β , c 2 B 1 β α 1 ( σ ( t ) , T ) , c 2  are any positive constant , if  α > β , g 1 ( t , T , c 1 , c 2 ) = Φ Δ ( t ) + 2 β Φ ( t ) τ Δ ( t ) h 2 ( τ ( t ) , T ) δ 1 ( τ ( t ) , T , c 1 , c 2 ) ( a n ( t ) ϕ ( t ) ) σ , g 2 ( t , T , c 1 , c 2 ) = Φ Δ ( t ) + ( 1 + 1 α ) β Φ ( t ) τ Δ ( t ) h 1 ( τ ( t ) , T ) g 2 ( t , T , c 1 , c 2 ) = × δ 2 ( τ ( t ) , T , c 1 , c 2 ) ( ( a n ( t ) ϕ ( t ) ) σ ) 1 α , G 1 ( t , T , c 1 , c 2 ) = Φ ( t ) q ( t ) Φ ( t ) ( a n ( t ) ϕ ( t ) ) Δ + β Φ ( t ) τ Δ ( t ) h 2 ( τ ( t ) , T ) G 1 ( t , T , c 1 , c 2 ) = × δ 1 ( τ ( t ) , T , c 1 , c 2 ) ( ( a n ( t ) ϕ ( t ) ) σ ) 2 , G 2 ( t , T , c 1 , c 2 ) = Φ ( t ) q ( t ) Φ ( t ) ( a n ( t ) ϕ ( t ) ) Δ + β α Φ ( t ) τ Δ ( t ) h 1 ( τ ( t ) , T ) G 2 ( t , T , c 1 , c 2 ) = × δ 2 ( τ ( t ) , T , c 1 , c 2 ) ( ( a n ( t ) ϕ ( t ) ) σ ) 1 + 1 α , g + = max { 0 , g } , g = min { 0 , g } , X ( t ) = { a n ( t ) [ [ S n 1 Δ ( t , x ( t ) ) ] α x β ( τ ( t ) ) + ϕ ( t ) ] } σ .
(3.2)

Theorem 3.1 Suppose that (2.1) and (2.2) hold. If there exist differentiable functions Φ:T(0,) and ϕ:T[0,) with a n (t)ϕ(t) being differentiable such that for all sufficiently large T [ t 0 , ) T and for any positive constants c 1 , c 2 , there is a T 1 >T with τ( T 1 )>T such that

lim sup t T 1 t [ G 1 ( s , T , c 1 , c 2 ) g 1 2 ( s , T , c 1 , c 2 ) 4 β Φ ( s ) τ Δ ( s ) h 2 ( τ ( s ) , T ) δ 1 ( τ ( s ) , T , c 1 , c 2 ) ] Δs=,
(3.3)

then every solution of (1.1) is either oscillatory or tends to 0.

Proof Assume that (1.1) has a nonoscillatory solution x on [ t 0 , ) T . Then, without loss of generality, there is a sufficiently large t 1 t 0 such that x(t)>0 for t t 1 . Therefore from Lemma 2.3, we know that there exists sufficiently large T t 1 such that:

  1. (1)

    S n Δ (t,x(t))<0 for tT.

  2. (2)

    Either lim t x(t)=0 or S i (t,x(t))>0 for tT and 0in.

Let S i (t,x(t))>0 for tT and 0in. Consider

w(t)=Φ(t) a n (t) [ ( S n 1 Δ ( t , x ( t ) ) ) α x β ( τ ( t ) ) + ϕ ( t ) ] .
(3.4)

Then X(t)= w σ (t)/ Φ σ (t) and w(t)>0 for tT.

By the product rule and the quotient rule

w Δ ( t ) = Φ ( t ) x β ( τ ( t ) ) S n Δ ( t , x ( t ) ) + [ Φ ( t ) x β ( τ ( t ) ) ] Δ S n σ ( t , x ( t ) ) + Φ ( t ) [ a n ( t ) ϕ ( t ) ] Δ + Φ Δ ( t ) [ a n ( t ) ϕ ( t ) ] σ .

Since g(t,x(τ(t)))/ x β (τ(t))q(t) (x(t)>0), we get

w Δ ( t ) Φ ( t ) q ( t ) + Φ ( t ) [ a n ( t ) ϕ ( t ) ] Δ + Φ Δ ( t ) X ( t ) Φ ( t ) [ ( x β ( τ ( t ) ) ) Δ x β ( τ ( t ) ) ] [ S n ( t , x ( t ) ) x β ( τ ( t ) ) ] σ .
(3.5)

Using the fact that x and τ are differentiable functions and τσ=στ, we see that xτ is a differentiable function and ( x ( τ ( t ) ) ) Δ = x Δ (τ(t)) τ Δ (t). Note β1. From Lemma 2.5, we get

( x β ( τ ( t ) ) ) Δ β x β 1 ( τ ( t ) ) x Δ ( τ ( t ) ) τ Δ (t),

which implies

w Δ ( t ) Φ ( t ) q ( t ) + Φ ( t ) [ a n ( t ) ϕ ( t ) ] Δ + Φ Δ ( t ) X ( t ) β Φ ( t ) τ Δ ( t ) x Δ ( τ ( t ) ) x ( τ ( t ) ) [ S n ( t , x ( t ) ) x β ( τ ( t ) ) ] σ .
(3.6)

We choose t 2 T such that τ(t)>T for t t 2 . Then, from (2.3) and the fact that S n Δ (t,x(t))<0 for tT, we get

x Δ ( τ ( t ) ) S n 1 α ( τ ( t ) , x ( t ) ) B 2 ( τ ( t ) , T ) a 1 ( τ ( t ) ) [ S n 1 α ( t , x ( t ) ) ] σ h 1 ( τ ( t ) , T ) = h 1 ( τ ( t ) , T ) ( a n 1 α ( t ) S n 1 Δ ( t , x ( t ) ) ) σ
(3.7)
= h 1 ( τ ( t ) , T ) [ S n ( t , x ( t ) ) x β ( τ ( t ) ) ] σ [ x β ( τ ( t ) ) S n α 1 α ( τ ( t , x ( t ) ) ) ] σ .
(3.8)

From (2.3), we have

x(t) S n 1 α ( t , x ( t ) ) B 1 (t,T).

Thus

S n α 1 α ( τ ( t ) , x ( t ) ) x α 1 ( τ ( t ) ) B 1 α 1 ( τ ( t ) , T ) ,

which combines with (3.8) to imply

x Δ ( τ ( t ) ) h 2 ( τ ( t ) , T ) [ ( S n ( t , x ( t ) ) ) x β ( τ ( t ) ) ] σ [ x β ( τ ( t ) ) x α 1 ( τ ( t ) ) ] σ .
(3.9)

Combining (3.9) with (3.6) and from x σ ( τ ( t ) ) x ( τ ( t ) ) 1, we obtain that

w Δ ( t ) Φ ( t ) q ( t ) + Φ ( t ) [ a n ( t ) ϕ ( t ) ] Δ + Φ Δ ( t ) X ( t ) β Φ ( t ) τ Δ ( t ) h 2 ( τ ( t ) , T ) [ x β ( τ ( t ) ) x α ( τ ( t ) ) ] σ { [ S n ( t , x ( t ) ) x β ( τ ( t ) ) ] σ } 2 .

Now we consider the following three cases.

Case (i). If α<β, then x Δ (t)>0 for tT and x(t)x(T)= b 1 >0. Thus

( x β α ( τ ( t ) ) ) σ b 1 β α = c 1 >0for t t 2 .

Case (ii). If α=β, then

( x β α ( τ ( t ) ) ) σ =1for t t 2 .

Case (iii). If α>β, then from (2.4) we get that there exist t 3 > t 2 and a constant c>0 such that

x(t)c B 1 (t,T)for t t 3 .

Thus

( x β α ( τ ( t ) ) ) σ c 2 ( B 1 β α ( τ ( t ) , T ) ) σ for t t 3 ,

where c 2 = c β α >0.

We obtain from the above that

( x β ( τ ( t ) ) x α ( τ ( t ) ) ) σ δ 1 ( τ ( t ) , T , c 1 , c 2 ) .

Thus

w Δ ( t ) Φ ( t ) q ( t ) + Φ ( t ) [ a n ( t ) ϕ ( t ) ] Δ + Φ Δ ( t ) X ( t ) β Φ ( t ) τ Δ ( t ) h 2 ( τ ( t ) , T ) δ 1 ( τ ( t ) , T , c 1 , c 2 ) { [ S n ( t , x ( t ) ) x β ( τ ( t ) ) ] σ } 2 .
(3.10)

Since

[ ( S n ( t , x ( t ) ) x β ( τ ( t ) ) ) σ ] 2 = ( X ( t ) ( a n ( t ) ϕ ( t ) ) σ ) 2 = X 2 ( t ) 2 ( a n ( t ) ϕ ( t ) ) σ X ( t ) + ( ( a n ( t ) ϕ ( t ) ) σ ) 2 ,
(3.11)

from (3.10) and (3.11) and the definitions of G 1 (t,T, c 1 , c 2 ) and g 1 (t,T, c 1 , c 2 ), we get

w Δ ( t ) G 1 ( t , T , c 1 , c 2 ) + g 1 ( t , T , c 1 , c 2 ) X ( t ) β Φ ( t ) τ Δ ( t ) h 2 ( τ ( t ) , T ) δ 1 ( τ ( t ) , T , c 1 , c 2 ) X 2 ( t ) .
(3.12)

It is easy to check that

w Δ (t) G 1 (t,T, c 1 , c 2 )+ g 1 2 ( t , T , c 1 , c 2 ) 4 β Φ ( t ) τ Δ ( t ) h 2 ( τ ( t ) , T ) δ 1 ( τ ( t ) , T , c 1 , c 2 ) .

Integrating both sides of the above inequality from t 3 to t, we get

t 3 t [ G 1 ( s , T , c 1 , c 2 ) g 1 2 ( s , T , c 1 , c 2 ) 4 β Φ ( s ) τ Δ ( s ) h 2 ( τ ( s ) , T ) δ 1 ( τ ( s ) , T , c 1 , c 2 ) ] Δ s w ( t 3 ) w ( t ) w ( t 3 ) ,

which leads to a contradiction to (3.3). The proof is completed. □

Theorem 3.2 Suppose that (2.1) and (2.2) hold. If there exist differentiable functions Φ:T(0,) and ϕ:T[0,) with a n (t)ϕ(t) being differentiable such that for all sufficiently large T [ t 0 , ) T and for any positive constants c 1 , c 2 , there is a T 1 >T with τ( T 1 )>T such that

lim sup t T 1 t [ G 2 ( s , T , c 1 , c 2 ) α α ( ( g 2 ( s , T , c 1 , c 2 ) ) + ) 1 + α ( 1 + α ) 1 + α ( β Φ ( s ) τ Δ ( s ) h 1 ( τ ( s ) , T ) δ 2 ( τ ( s ) , T , c 1 , c 2 ) ) α ] Δ s = ,
(3.13)

then every solution of (1.1) is either oscillatory or tends to 0.

Proof Assume that (1.1) has a nonoscillatory solution x on [ t 0 , ) T . Then, without loss of generality, there is a sufficiently large t 1 t 0 such that x(t)>0 for t t 1 . Therefore from Lemma 2.3, we know that there exists sufficiently large T t 1 such that:

  1. (1)

    S n Δ (t,x(t))<0 for tT.

  2. (2)

    Either lim t x(t)=0 or S i (t,x(t))>0 for tT and 0in.

Let S i (t,x(t))>0 for tT and 0in. From (3.7), we get

x Δ ( τ ( t ) ) [ S n 1 α ( t , x ( t ) ) ] σ h 1 ( τ ( t ) , T ) = h 1 ( τ ( t ) , T ) ( x β α ( τ ( t ) ) ) σ { [ S n ( t , x ( t ) ) x β ( τ ( t ) ) ] σ } 1 α .
(3.14)

Define w(t) as (3.4). Choosing t 2 T such that τ(t)>T for t t 2 . Combining (3.14) with (3.6), we see that for t [ t 2 , ) T ,

w Δ ( t ) Φ ( t ) q ( t ) + Φ ( t ) [ a n ( t ) ϕ ( t ) ] Δ + Φ Δ ( t ) X ( t ) β Φ ( t ) τ Δ ( t ) h 1 ( τ ( t ) , T ) { [ S n ( t , x ( t ) ) x β ( τ ( t ) ) ] σ } 1 + 1 α ( x β α ( τ ( t ) ) ) σ x ( τ ( t ) ) .

Note x(τ(t)) ( x ( τ ( t ) ) ) σ since x Δ (t)>0, we obtain

w Δ ( t ) Φ ( t ) q ( t ) + Φ ( t ) [ a n ( t ) ϕ ( t ) ] Δ + Φ Δ ( t ) X ( t ) β Φ ( t ) τ Δ ( t ) h 1 ( τ ( t ) , T ) ( ( S n ( t , x ( t ) ) x β ( τ ( t ) ) ) σ ) 1 + 1 α ( x β α α ( τ ( t ) ) ) σ .

Now we consider the following three cases.

Case (i). If α<β, then

( x β α α ( τ ( t ) ) ) σ x β α α (T)= c 1 >0for t t 2

since x Δ (t)>0 for tT.

Case (ii). If α=β, then

( x β α α ( τ ( t ) ) ) σ =1for t t 2 .

Case (iii). If α>β, then we get from (2.4) that there exist t 3 > t 2 and a constant c>0 such that

x ( τ ( t ) ) c B 1 ( τ ( t ) , T ) for t t 3 .

Thus

( x β α α ( τ ( t ) ) ) σ c 2 ( B 1 β α α ( τ ( t ) , T ) ) σ for t t 3 ,

where c 2 = c β α α >0.

We obtain from the above

( x β α α ( τ ( t ) ) ) σ δ 2 ( τ ( t ) , T , c 1 , c 2 ) .

Then

w Δ ( t ) Φ ( t ) q ( t ) + Φ ( t ) [ a n ( t ) ϕ ( t ) ] Δ + Φ Δ ( t ) X ( t ) β Φ ( t ) τ Δ ( t ) h 1 ( τ ( t ) , T ) δ 2 ( τ ( t ) , T , c 1 , c 2 ) { [ S n ( t , x ( t ) ) x β ( τ ( t ) ) ] σ } 1 + 1 α .
(3.15)

From Lemma 2.7, we have

( ( S n ( t , x ( t ) ) x β ( τ ( t ) ) ) σ ) 1 + 1 α = ( X ( t ) ( a n ( t ) ϕ ( t ) ) σ ) 1 + 1 α X 1 + 1 α ( t ) + 1 α ( a n σ ( t ) ϕ σ ( t ) ) 1 + 1 α ( 1 + 1 α ) ( a n σ ( t ) ϕ σ ( t ) ) 1 α X ( t ) .
(3.16)

Combining (3.16) with (3.15) and the definitions of G 2 (t,T, c 1 , c 2 ) and g 2 (t,T, c 1 , c 2 ), we get

w Δ ( t ) G 2 ( t , T , c 1 , c 2 ) + g 2 ( t , T , c 1 , c 2 ) X ( t ) β Φ ( t ) τ Δ ( t ) h 1 ( τ ( t ) , T ) δ 2 ( τ ( t ) , T , c 1 , c 2 ) X 1 + 1 α ( t ) G 2 ( t , T , c 1 , c 2 ) + ( g 2 ( t , T , c 1 , c 2 ) ) + X ( t ) β Φ ( t ) τ Δ ( t ) h 1 ( τ ( t ) , T ) δ 2 ( τ ( t ) , T , c 1 , c 2 ) X 1 + 1 α ( t ) .
(3.17)

Let

A 1 + 1 α =βΦ(t) τ Δ (t) h 1 ( τ ( t ) , T ) δ 2 ( τ ( t ) , T , c 1 , c 2 ) X 1 + 1 α (t),
(3.18)
B 1 α = α ( g 2 ( t , T , c 1 , c 2 ) ) + ( 1 + α ) ( β Φ ( t ) τ Δ ( t ) h 1 ( τ ( t ) , T ) δ 2 ( τ ( t ) , T , c 1 , c 2 ) ) α α + 1 .
(3.19)

We have from Lemma 2.6

( g 2 ( t , T , c 1 , c 2 ) ) + X ( t ) β Φ ( t ) τ Δ ( t ) h 1 ( τ ( t ) , T ) δ 2 ( τ ( t ) , T , c 1 , c 2 ) X 1 + 1 α ( t ) α α ( ( g 2 ( t , T , c 1 , c 2 ) ) + ) 1 + α ( 1 + α ) 1 + α ( β Φ ( t ) τ Δ ( t ) h 1 ( τ ( t ) , T ) δ 2 ( τ ( t ) , T , c 1 , c 2 ) ) α .

Then

w Δ (t) G 2 (t,T, c 1 , c 2 )+ α α ( ( g 2 ( t , T , c 1 , c 2 ) ) + ) 1 + α ( 1 + α ) 1 + α ( β Φ ( t ) τ Δ ( t ) h 1 ( τ ( t ) , T ) δ 2 ( τ ( t ) , T , c 1 , c 2 ) ) α .

Integrating both sides of the above inequality from t 3 to t, we get

t 3 t [ G 2 ( s , T , c 1 , c 2 ) α α ( ( g 2 ( s , T , c 1 , c 2 ) ) + ) 1 + α ( 1 + α ) 1 + α ( β Φ ( s ) τ Δ ( s ) h 1 ( τ ( s ) , T ) δ 2 ( τ ( s ) , T , c 1 , c 2 ) ) α ] Δ s w ( t 3 ) w ( t ) w ( t 3 ) ,

which leads to a contradiction to (3.13). The proof is completed. □

4 Further results

For convenience, let D={(t,s) T 2 :ts t 0 ,t,s [ t 0 , ) T }. For any function G: T 2 R, denote by G Δ s the partial derivative of G(t,s) with respect to s. Define

= { G C rd ( D , [ 0 , ) ) : G ( s , s ) = 0 , G ( t , s ) > 0 , G Δ s 0 , t > s t 0 } .

Theorem 4.1 Suppose that (2.1) and (2.2) hold. If there exist functions r,R and differentiable functions Φ:T(0,) and ϕ:T[0,) with a n (t)ϕ(t) being differentiable such that for all sufficiently large T [ t 0 , ) T and for any positive constants c 1 , c 2 , there is a T 1 >T with τ( T 1 )>T such that

R Δ s (t,s)+ R ( t , s ) g 1 ( s , T , c 1 , c 2 ) Φ σ ( s ) = r ( t , s ) Φ σ ( s ) R 1 2 (t,s)
(4.1)

and

lim sup t 1 R ( t , T 1 ) T 1 t [ R ( t , s ) G 1 ( s , T , c 1 , c 2 ) r 2 ( t , s ) 4 β Φ ( s ) τ Δ ( s ) h 2 ( τ ( s ) , T ) δ 1 ( τ ( s ) , T , c 1 , c 2 ) ] Δ s = ,
(4.2)

then every solution of (1.1) is either oscillatory or tends to 0.

Proof Assume that (1.1) has a nonoscillatory solution x on [ t 0 , ) T . Then, without loss of generality, there is a sufficiently large t 1 t 0 such that x(t)>0 for t t 1 . Therefore from Lemma 2.3, we know that there exists sufficiently large T t 1 such that:

  1. (1)

    S n Δ (t,x(t))<0 for tT.

  2. (2)

    Either lim t x(t)=0 or S i (t,x(t))>0 for tT and 0in.

Let S i (t,x(t))>0 for tT and 0in. Define w(t) as (3.4). Choosing t 2 T such that (3.12) holds for t t 2 . Then for t [ t 2 , ) T

G 1 ( t , T , c 1 , c 2 ) w Δ ( t ) + g 1 ( t , T , c 1 , c 2 ) X ( t ) β Φ ( t ) τ Δ ( t ) h 2 ( τ ( t ) , T ) δ 1 ( τ ( t ) , T , c 1 , c 2 ) X 2 ( t ) .
(4.3)

In (4.3), replace t by s and multiply both sides by R(t,s), integrate with respect to s from t 2 to t> t 2 , we have

t 2 t R ( t , s ) G 1 ( s , T , c 1 , c 2 ) Δ s t 2 t R ( t , s ) w Δ ( s ) Δ s + t 2 t R ( t , s ) g 1 ( s , T , c 1 , c 2 ) X ( s ) Δ s t 2 t R ( t , s ) β Φ ( s ) τ Δ ( s ) h 2 ( τ ( s ) , T ) δ 1 ( τ ( s ) , T , c 1 , c 2 ) X 2 ( s ) Δ s .

Integrating by parts and using (4.1), we get

t 2 t R ( t , s ) G 1 ( s , T , c 1 , c 2 ) Δ s R ( t , t 2 ) w ( t 2 ) + t 2 t [ r ( t , s ) R 1 2 ( t , s ) X ( s ) R ( t , s ) β Φ ( s ) τ Δ ( s ) h 2 ( τ ( s ) , T ) δ 1 ( τ ( s ) , T , c 1 , c 2 ) X 2 ( s ) ] Δ s .

This implies

t 2 t R ( t , s ) G 1 ( s , T , c 1 , c 2 ) Δ s R ( t , t 2 ) w ( t 2 ) + t 2 t r 2 ( t , s ) 4 β Φ ( s ) τ Δ ( s ) h 2 ( τ ( s ) , T ) δ 1 ( τ ( s ) , T , c 1 , c 2 ) Δ s .

Thus

1 R ( t , t 2 ) t 2 t [ R ( t , s ) G 1 ( s , T ) r 2 ( t , s ) 4 β Φ ( s ) τ Δ ( s ) h 2 ( τ ( s ) , T ) δ 1 ( τ ( s ) , T , c 1 , c 2 ) ] Δsw( t 2 ),

which leads to a contradiction to (4.2). The proof is completed. □

Theorem 4.2 Suppose that (2.1) and (2.2) hold. If there exist functions r,R and differentiable functions Φ:T(0,) and ϕ:T[0,) with a n (t)ϕ(t) being differentiable such that for all sufficiently large T [ t 0 , ) T and for any positive constants c 1 , c 2 , there is a T 1 >T with τ( T 1 )>T such that

R Δ s (t,s)+ R ( t , s ) g 2 ( s , T , c 1 , c 2 ) Φ σ ( s ) = r ( t , s ) Φ σ ( s ) R α 1 + α (t,s)
(4.4)

and

lim sup t 1 R ( t , T 1 ) T 1 t [ R ( t , s ) G 2 ( s , T , c 1 , c 2 ) α α ( r 2 ( t , s ) ) 1 + α ( 1 + α ) 1 + α ( β Φ ( s ) τ Δ ( s ) h 2 ( τ ( s ) , T ) δ 1 ( τ ( s ) , T , c 1 , c 2 ) ) α ] Δ s = ,
(4.5)

then every solution of (1.1) is either oscillatory or tends to 0.

Proof Assume that (1.1) has a nonoscillatory solution x on [ t 0 , ) T . Then, without loss of generality, there is a sufficiently large t 1 t 0 such that x(t)>0 for t t 1 . Therefore from Lemma 2.3, we know that there exists sufficiently large T t 1 such that:

  1. (1)

    S n Δ (t,x(t))<0 for tT.

  2. (2)

    Either lim t x(t)=0 or S i (t,x(t))>0 for tT and 0in.

Let S i (t,x(t))>0 for tT and 0in. Define w(t) as (3.4). Choosing t 2 T such that (3.17) holds for t t 2 . Then for t [ t 2 , ) T , we have

G 2 ( t , T , c 1 , c 2 ) w Δ ( t ) + g 2 ( t , T , c 1 , c 2 ) Φ σ ( t ) w σ ( t ) β Φ ( t ) τ Δ ( t ) h 1 ( τ ( t ) , T ) δ 1 ( τ ( t ) , T , c 1 , c 2 ) X ( 1 + 1 α ) ( t ) .
(4.6)

In (4.6), replace t by s and multiply both sides by R(t,s) and integrate with respect to s from t 2 to t> t 2 , it follows that

t 2 t R ( t , s ) G 2 ( s , T , c 1 , c 2 ) Δ s t 2 t R ( t , s ) w Δ ( s ) Δ s + t 2 t R ( t , s ) g 2 ( s , T , c 1 , c 2 ) Φ σ ( s ) w σ ( s ) Δ s t 2 t R ( t , s ) β Φ ( s ) τ Δ ( s ) h 1 ( τ ( s ) , T ) δ 2 ( τ ( s ) , T , c 1 , c 2 ) X ( 1 + 1 α ) ( s ) Δ s .

Integrating by parts and using (4.4), we get

t 2 t R ( t , s ) G 2 ( s , T , c 1 , c 2 ) Δ s R ( t , t 2 ) w ( t 2 ) + t 2 t [ r + ( t , s ) R α 1 + α ( t , s ) X ( s ) β R ( t , s ) Φ ( s ) τ Δ ( s ) h 1 ( τ ( s ) , T ) δ 2 ( τ ( s ) , T , c 1 , c 2 ) X ( 1 + 1 α ) ( s ) ] Δ s .
(4.7)

Let

A 1 + 1 α =βR(t,s)Φ(s) τ Δ (s) h 1 ( τ ( s ) , T ) δ 2 ( τ ( s ) , T , c 1 , c 2 ) X 1 + 1 α (s),
(4.8)
B 1 α = α r + ( t , s ) ( 1 + α ) ( β Φ ( s ) τ Δ ( s ) h 1 ( τ ( s ) , T ) δ 2 ( τ ( s ) , T , c 1 , c 2 ) ) α α + 1 .
(4.9)

From Lemma 2.6, we have

t 2 t [ r + ( t , s ) R α 1 + α ( t , s ) X ( s ) β R ( t , s ) Φ ( s ) τ Δ ( s ) h 1 ( τ ( s ) , T ) δ 2 ( τ ( s ) , T , c 1 , c 2 ) X 1 + 1 α ( s ) ] Δ s t 2 t α α ( r + ( t , s ) ) 1 + α ( 1 + α ) 1 + α ( β Φ ( s ) τ Δ ( s ) h 1 ( τ ( s ) , T ) δ 2 ( τ ( s ) , T , c 1 , c 2 ) ) α Δ s ,

which implies

t 2 t [ R ( t , s ) G 2 ( s , T , c 1 , c 2 ) α α ( r + ( t , s ) ) 1 + α ( 1 + α ) 1 + α ( β Φ ( s ) τ Δ ( s ) h 1 ( τ ( s ) , T ) δ 2 ( τ ( s ) , T , c 1 , c 2 ) ) α ] Δ s R ( t , t 2 ) w ( t 2 ) .

Then

1 R ( t , t 2 ) t 2 t [ R ( t , s ) G 2 ( s , T , c 1 , c 2 ) α α ( r + ( t , s ) ) 1 + α ( 1 + α ) 1 + α ( β Φ ( s ) τ Δ ( t ) h 1 ( τ ( s ) , T ) δ 2 ( τ ( s ) , T , c 1 , c 2 ) ) α ] Δ s w ( t 2 ) ,

which leads to a contradiction to (4.2). The proof is completed. □

5 Example

In this section, we give an example to illustrate our main results.

Example 5.1 Consider the following higher order dynamic equation:

S n Δ ( t , x ( t ) ) + γ t β + 1 x β + 1 ( τ ( t ) ) =0
(5.1)

on time scale T={0}{1/ 2 k :k=1,2,3,}{ 2 k :k=1,2,3,}, where n2, S k (t) (0kn) is as in (1.1) with a n (t)= t α , a n 1 (t)== a 1 (t)=1, q(t)= γ t β + 1 , γ>1/ 2 1 β , 0<β<1, t 0 =4, and τ(t)=t/2. Then τ Δ (t)=1/2 and the forward jump operator σ(t)=2t satisfies σ(τ(t))=τ(σ(t)). Thus

t 0 ( 1 a n ( s ) ) 1 α Δ s = t 0 Δ s s = , t 0 Δ s a i ( s ) = t 0 Δ s = , t 0 1 a n 1 ( t ) { t [ 1 a n ( s ) s q ( u ) Δ u ] 1 α Δ s } Δ t = t 0 { t [ 1 s α s γ u β + 1 Δ u ] 1 α Δ s } Δ t ( γ β ) 1 α t 0 { t [ 1 s α s ( u β ) Δ u β ( u β ) σ Δ u ] 1 α Δ s } Δ t = ( γ β ) 1 α t 0 [ t Δ s s s β α ] Δ t ( γ β ) 1 α t 0 [ t Δ s s s σ ] Δ t = ( γ β ) 1 α t 0 1 t Δ t = .

Therefore (2.1) and (2.2) hold. Note that

lim t B n (t,T)= lim t T t [ 1 a n ( s ) ] 1 α Δs= lim t T t 1 s Δs=.

It is easy to check that

lim t B 2 (t,T)= lim t B 1 ( σ ( t ) , T ) =

and

lim t h 2 (t,T)= lim t h 1 ( σ ( t ) , T ) =.

Then for any positive constants c 1 , c 2 , there is a sufficiently large t 1 such that ( B 1 ( τ ( t ) , T ) ) σ >1 for t t 1 , h 1 (τ(t),T)1/ c 2 , and h 2 (τ(t),T)max{1,1/ c 1 } for t t 1 ,

h 2 ( τ ( t ) , T ) δ 1 ( τ ( t ) , T , c 1 , c 2 ) = { c 1 h 2 ( τ ( t ) , T ) 1 , if  α < β + 1 , h 2 ( τ ( t ) , T ) 1 , if  α = β + 1 , c 2 h 1 ( τ ( t ) , T ) ( B 1 β ( τ ( t ) , T ) ) σ 1 , if  α > β + 1 .

Choosing ϕ(t)=0 and Φ(t)=t. Then g 1 (t,T, c 1 , c 2 )= Φ Δ (t)=1 and G 1 (t,T, c 1 , c 2 )=Φ(t)q(t)= γ t β . Thus

lim sup t T 1 t [ G 1 ( s , T , c 1 , c 2 ) g 1 2 ( s , T , c 1 , c 2 ) 4 ( β + 1 ) Φ ( s ) τ Δ ( s ) h 2 ( τ ( s ) , T ) δ 1 ( τ ( s ) , T , c 1 , c 2 ) ] Δ s lim sup t T 1 t ( γ s β 1 2 ( β + 1 ) 1 s ) Δ s lim sup t T 1 t 1 2 s Δ s = .

The conditions of Theorem 3.1 are satisfied. Then every solution of (5.1) is either oscillatory or tends to 0.

Remark 5.1 If β=0, then the conditions of Theorem 3.1 are also satisfied and every solution of (5.1) is also either oscillatory or tends to 0.

Remark 5.2 In Example 5.1, let R(t,s)=r(t,s)=1 for t>s4 and R(t,t)=r(t,t)=0 for t4. Then the conditions of Theorem 4.1 are satisfied. It also follows from Theorem 4.1 that every solution of (5.1) is either oscillatory or tends to 0.

References

  1. 1.

    Hilger S: Analysis on measure chains - a unified approach to continuous and discrete calculus. Results Math. 1990, 18: 18-56. 10.1007/BF03323153

  2. 2.

    Kac V, Chueng P Universitext. In Quantum Calculus. Springer, New York; 2002.

  3. 3.

    Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston; 2001.

  4. 4.

    Bohner M, Peterson A: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston; 2003.

  5. 5.

    Hassan TS: Oscillation of third order nonlinear delay dynamic equations on time scales. Math. Comput. Model. 2009, 49: 1573-1586. 10.1016/j.mcm.2008.12.011

  6. 6.

    Erbe L, Peterson A, Saker SH: Asymptotic behavior of solutions of a third-order nonlinear dynamic equation on time scales. J. Comput. Appl. Math. 2005, 181: 92-102. 10.1016/j.cam.2004.11.021

  7. 7.

    Erbe L, Peterson A, Saker SH: Oscillation and asymptotic behavior a third-order nonlinear dynamic equation. Can. Appl. Math. Q. 2006, 14: 129-147.

  8. 8.

    Erbe L, Peterson A, Saker SH: Hille and Nehari type criteria for third-order dynamic equations. J. Math. Anal. Appl. 2007, 329: 112-131. 10.1016/j.jmaa.2006.06.033

  9. 9.

    Li T, Han Z, Sun S, Zhao Y: Oscillation results for third order nonlinear delay dynamic equations on time scales. Bull. Malays. Math. Soc. 2011, 34: 639-648.

  10. 10.

    Wang Y, Xu Z: Asymptotic properties of solutions of certain third-order dynamic equations. J. Comput. Appl. Math. 2012, 236: 2354-2366. 10.1016/j.cam.2011.11.021

  11. 11.

    Erbe L, Hassan TS, Peterson A: Oscillation of third order nonlinear functional dynamic equations on time scales. Differ. Equ. Dyn. Syst. 2008, 18: 199-227.

  12. 12.

    Qi Y, Yu J: Oscillation criteria for fourth-order nonlinear delay dynamic equations. Electron. J. Differ. Equ. 2013., 2013: Article ID 79

  13. 13.

    Grace SR, Bohner M, Sun S: Oscillation of fourth-order dynamic equations. Hacet. J. Math. Stat. 2010, 39: 545-553.

  14. 14.

    Grace SR, Sun S, Wang Y: On the oscillation of fourth order strongly superlinear and strongly sublinear dynamic equations. J. Appl. Math. Comput. 2014, 44: 119-132. 10.1007/s12190-013-0684-y

  15. 15.

    Agarwal RP, Bohner M, Li T, Zhang C: Oscillation theorems for fourth-order half-linear delay dynamic equations with damping. Mediterr. J. Math. 2014, 11: 463-475. 10.1007/s00009-013-0310-0

  16. 16.

    Zhang C, Li T, Agarwal RP, Bohner M: Oscillation results for fourth-order nonlinear dynamic equations. Appl. Math. Lett. 2012, 25: 2058-2065. 10.1016/j.aml.2012.04.018

  17. 17.

    Sun T, He Q, Xi H, Yu W: Oscillation for higher order dynamic equations on time scales. Abstr. Appl. Anal. 2013., 2013: Article ID 268721

  18. 18.

    Agarwal RP, Bohner M, Li T, Zhang C: Oscillation criteria for second-order dynamic equations on time scales. Appl. Math. Lett. 2014, 31: 34-40.

  19. 19.

    Grace SR, Hassan TS: Oscillation criteria for higher order nonlinear dynamic equations. Math. Nachr. 2014, 287: 1659-1673. 10.1002/mana.201300157

  20. 20.

    Graef JR, Panigrahi S, Reddy PR: On oscillatory and asymptotic behavior of fourth order nonlinear neutral delay dynamic equations with positive and negative coefficients. Math. Slovaca 2014, 64: 347-366. 10.2478/s12175-014-0209-7

  21. 21.

    Li L, Han M: Some new dynamic Opial type inequalities and applications for second order integro-differential dynamic equations on time scales. Appl. Math. Comput. 2014, 232: 542-547.

  22. 22.

    Sun T, Xi H, Pen X, Yu W: Nonoscillatory solutions for higher-order neutral dynamic equations on time scales. Abstr. Appl. Anal. 2010., 2010: Article ID 428963

  23. 23.

    Sun T, Xi H, Yu W: Asymptotic behaviors of higher order nonlinear dynamic equations on time scales. J. Appl. Math. Comput. 2011, 37: 177-192. 10.1007/s12190-010-0428-1

  24. 24.

    Sun T, Xi H, Pen X: Asymptotic behavior of solutions of higher-order dynamic equations on time scales. Adv. Differ. Equ. 2011., 2011: Article ID 237219

  25. 25.

    Wu X, Sun T, Xi H, Chen C: Kamenev-type oscillation criteria for higher-order nonlinear dynamic equations on time scales. Adv. Differ. Equ. 2013., 2013: Article ID 248

  26. 26.

    Wu X, Sun T, Xi H, Chen C: Oscillation criteria for fourth-order nonlinear dynamic equations on time scales. Abstr. Appl. Anal. 2013., 2013: Article ID 740568

  27. 27.

    Wu, X, Sun, T: Oscillation criteria for higher order nonlinear delay dynamic equations on time scales. Math. Slovaca (2014, in press)

  28. 28.

    Sun T, Yu W, Xi H: Oscillatory behavior and comparison for higher order nonlinear dynamic equations on time scales. J. Appl. Math. Inform. 2012, 30: 289-304.

  29. 29.

    Gera M, Graef JR, Gregus M: On oscillatory and asymptotic properties of solutions of certain nonlinear third order differential equations. Nonlinear Anal., Theory Methods Appl. 1998, 32: 417-425. 10.1016/S0362-546X(97)00483-5

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Acknowledgements

This project is supported by NNSF of China (11461003) and NSF of Guangxi (2012GXNSFDA276040).

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Correspondence to Taixiang Sun.

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Keywords

  • oscillation
  • dynamic equation
  • time scale